The Ontological Import of Adding Proper Classes

Alfredo Roque Freire1 https://orcid.org/0000-0003-0132-355X 1University of Campinas Department of Philosophy Campinas, SP Brazil e-mail: [email protected]

Rodrigo de Alvarenga Freire2 https://orcid.org/0000-0002-5324-4645 2University of Brasília Department of Philosophy Brasília, DF Brazil e-mail: [email protected]

Article Info: CDD: 511.3 Received: 18.01.2019; Revised: 25.01.2019; Accepted: 02.02.2019 DOI: http://dx.doi.org/10.1590/0100-6045.2019.V42N2.RF

Keywords: Set theory Class theory Formal ontology

Abstract: In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to

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NBG. We thus consider the standard techniques of reduc- ing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of the same formula. We argue that the first makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. Moreover, we will modernize Shoenfield’s proof, emphasizing its rela- tion to Herbrand’s theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are pow- erful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological content.

1 How can we compare the ontological content of diﬀerent theories?

Within the context of a formal theory, an assertion is ontologically committing if it expresses a closure property of the intended models. For example, the power set and the union axioms express closure properties in set theories. If X and Y are elements in our set theory model, then the axioms guarantee that their union and ther power sets are also in the model. This property represents how an axiom generates ontological import. A detailed analysis of this conception leads to an un- derstanding of existential aspects in formal theories (see [4]). But it is of no help if we want to compare diﬀerent theories regarding ontology. We propose a diﬀerent approach for this kind of comparison: If T1 and T2 are formal theories and T2 can reduce T1 but T1 cannot reduce T2 using a natural method of ontological reduction given in a metatheory, then the ontological

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content of T2 is greater than that of T1. Now, what is a natural method of ontological reduction? We expect that if T1 reduces T2, then the consistency of T1 can be proved from the consistency of T2 in the corresponding metatheory. This and other related questions are dealt with in [1], which is the source of the main results in this paper. Nevertheless, we will not be concerned with this question in its full generality here, as we will concentrate on methods of interpretations between first-order theories in a finitary metatheory. More precisely, using the method of interpretations and the above conception of ontological comparison, we prove that the ontological content of NBG is greater than the ontological content of ZF. This result goes against the received view according to which those theories are equivalent. We will analyze the received view and point out its insufficiencies. It is based on the folklore conservativ- ity result: NBG is conservative with respect to ZF. A detailed finitary proof of this result will be provided, as there is basically no modern proof for this in the lit- erature. We will argue that the ontological reduction operating here is not conclusive, for the corresponding reduction method is partial. Hence, it is not clear if it gives a right transposition of the ontology. Also, it is always possible to strengthen the metatheory and weaken the reduction method in order to trivialize the comparison. Therefore, for the equivalence claim, it is not enough to have one mutual comparison according to some reduction method in some metatheory. In the opposite direction, it is very significant to have an asymmetric comparison by a standard method of reduction, and this result will be proved subsequently, together with other related results. The absence of interpretation from NBG to ZF thus understood is a strong evidence for the thesis that the ontological content of the former

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2 Novak’s model-theoretic reduction of NBG to ZF

From every model of ZF, a model of NBG can be obtained by the addition of definable proper classes. This construction has the important feature that the resulting model has the same sets as the original model. Therefore, the following reduction of NBG to ZF is obtained: Assume that α is a ZF-sentence and that NBG ` α. Let M be a model of ZF, and let N be the model of NBG obtained from M by the addition of the definable proper classes. Since NBG ` α, it follows that N |= α. However, if N |= α, then M |= α, for α is a ZF-sentence – it is about sets only – and M and N have the same sets. Now, the completeness theorem gives that ZF ` α, for M is an arbitrary model of ZF. The above reduction gives, in particular, a model- theoretic proof that if ZF is consistent, so is NBG. However, a bolder philosophical conclusion from this is that the ontological content of NBG is already present in ZF, as it can be easily fulfilled by the definable classes lurking in models of ZF. We claim that the bolder conclusion is unwarranted. The problem is that the metatheory in which the reduction takes place is too strong and the difference may be obliterated by its excessive strength. To make the point clear, assume that we were interested in comparing theories T1 and T2 in a metatheory which happens to be strong enough to prove the consistency of both theories. The equivalence between the consistency of T1 and that of T2 is, therefore, valid in such a metatheory, but no ontological comparison can be drawn. From a proof of equiconsistency in some metatheory, one cannot conclude ontological equivalence. How-

Manuscrito – Rev. Int. Fil. Campinas, v. 42, n. 2, pp. 85-112, Abr.-Jun. 2019. The Ontological Import of Adding Proper Classes 89 ever, there is another reduction of NBG to ZF pro- viding a finitary equiconsistency result due to Shoen- field. Although the above argument does not apply to Shoenfield’s reduction, which takes place in a finitary metatheory, we still claim that the bolder conclusion that the ontological content of NBG is already present in ZF is unwarranted. The problem is within the reduction method itself: It does not provide a reduction of the quantification over class variables, which gives NBG its extra ontological content, but rather just shows that the quantificational reasoning with class variables in NBG is dispensable and can be avoided in proofs of ZF-sentences. Therefore, there is no real reduction taking place here, there is no transposition of ontology, and the ontological equivalence does not follow. Shoenfield’s proof will be given in the next section and meticulously analyzed to support our claim.

3 Shoenﬁeld’s ﬁnitary reduction

The finitary proof of equiconsistency between NBG and ZF was provided by Shoenfield in the article "A relative consistency proof" ([9]). This article is written in the language of Principia Mathematica by White- head and Russell, and makes extensive use of the techniques developed in Grundlagen der Mathematik by Hilbert and Bernays. For this reason, we have developed this section by an excavation, a reverse en- gineering, in which the tools used were unraveled by the clues left in the article. In addition, changing the axiomatic system (see [7]) poses several additional dif- ficulties and, in many instances, the proof changes sig- nificantly. We will expose here the technique used for the equiconsistency proof. Before doing that, however, we need to remember the finitary proof technique of Herbrand’s theorem. For this, we will go through the necessary

Manuscrito – Rev. Int. Fil. Campinas, v. 42, n. 2, pp. 85-112, Abr.-Jun. 2019. Alfredo Roque Freire and Rodrigo de Alvarenga Freire 90 deﬁnitions, then we will enunciate Hilbert-Ackermann’s theorem and, ﬁnally, the necessary part of the strategy for establishing Herbrand’s theorem.

Deﬁnition 1 A formula α is open if all variables occurring in the formula are free.

Deﬁnition 2 A theory T is open if all its axioms are open formulas.

Deﬁnition 3 α is a quasi-tautology if, and only if, α is tautological consequence of instances of identity and equality axioms.

Theorem 4 (Hilbert-Ackermann) A open theory T is inconsistent if, and only if, there is a quasi-tautology α of the form ¬β1 ∨ ¬β2 ∨ ... ∨ ¬βk, such that βi is an instance of some axiom in T for each i ≤ k.

The ﬁnitary proof of this theorem can be found in [5, p. 48 - 52]. It is important to bear in mind this theorem since it is equivalent to the existential case of Herbrand’s theorem.

Deﬁnition 5 Let Q be a quantiﬁer ∀ or ∃, then a prenex formula is of the form:

Qx1Qx2 . . . Qxnθ, being θ a open formula. We call θ the matrix of Qx1Qx2 . . . Qxnθ.

We can write the prenex form, without loss in generality, with explicit quantiﬁers ∀ e ∃, instead of using the Q. The general formula is of the form:

∃z1∀y1 ... ∃zk∀yk θ[x, z1, y1,..., zk, yk], being x the sequence of free variables in the matrix θ. Using this notation will simplify the proof, since each quantiﬁer ∀ and ∃ is treated diﬀerently.

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Deﬁnition 6 A formula α is existential when α is a prenex formula of the form ∃x1∃x2 ... ∃xnθ, being θ an open formula.

From prenex formulas, we have the following theorem:

Theorem 7 For any formula α, there is a α0, such that:

1. α0 is a prenex formula;

2. ` α ↔ α0;

3. α0 is obtained by a primitive recursive procedure;

We call α0 the prenex form of α.

Next we will expose Herbrand’s normal form. It can be understood as the representation of any formula by an existential formula through a procedure of elimination of universal quantiﬁers. Such elimination is due to the introduction of function symbols in language.

Deﬁnition 8 (Herbrand’s normal form) Let α be any formula, we build αH through the following procedure:

1. α0 is the prenex form of α;

2. If αi is an existential formula, then αH is αi;

3. If αi if of the form ∃x1∃x2 ... ∃xn∀yγ, we introduce a function symbol f, such that:

αi+1 é ∃x1∃x2 ... ∃xnγy(f(x1, x2, . . . , xn)).

If αi is of the form ∀yγ, we introduce a constant symbol c, such that:

αi+1 é γy(c);

We can represent Herbrand’s normal form of a prenex formula

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∃z1∀y1 ... ∃zk∀yk θ[x, z1, y1,..., zk, yk], for

∃z1 ... ∃zk θ[x, z1, f1(z1),..., zk, fk(z1,..., zk)].

Theorem 9 (Herbrand) Let T be a theory without non logical axioms in the language L. Then, for any prenex formula α in L, it holds that: T ` α in the language L ⇐⇒ there is a quasi- tautology β1 ∨ β2 ... ∨ βk, for which, for each i, βi is an instance of the matrix αH .

Proof. (Detailed strategy) The procedure for proving Herbrand’s theorem follows the steps:

1. If α is an existential formula, the theorem is a corollary of Hilbert-Ackermann’s theorem:

We suppose that α is of the form ∃x1∃x2 ... ∃xnβ, with β as an open formula. In this case, ¬α is logically equivalent to ∀x1∀x2 ... ∀xn¬β. Thus, T ` ¬α ↔ ¬β. Because of that, T ` α ⇐⇒ the theory {¬β} is inconsistent. By the Hilbert-Ackermann’s theorem, for {¬β} is a open theory, {¬β} is inconsistent ⇐⇒ there is a quasi- tautology ¬(¬β1) ∨ ¬(¬β2) ∨ ... ∨ ¬(¬β1), with βi an instance of β for all i.

But this is equivalent to β1 ∨ β2 ∨ ... ∨ β1, ﬁnal- izing the proof.

2. We take, for the general case, a prenex formula α in the language L

∃z1∀y1 ... ∃zk∀yk θ[x, z1, y1,..., zk, yk].

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3. Let LH be the language L extended with functions used to built αH and TH the theory without logical axioms in the language LH , we should prove that:

T ` α ⇐⇒ TH ` αH .

4. Since αH is an existential formula, we obtain a quasi-tautology for the extended language LH. Hence,

T ` α ⇐⇒ TH ` αH if, and only if,

there is a quasi-tautology with instances of αH . For this reason, the proof of item 3 ﬁnish the proof.

5. To prove item 3, we only show the strategy for the converse implication, that is, T ` α ⇐ T 0 ` αH , since the direct proof is relatively simple.

6. Let Tc be the Henkin extension of T , deﬁned in [5, p. 46], and Tc+eq be the addition of equiva- 1 lence axioms of Herbrand [5, p. 52] in Tc, we show that:

(a) Tc is a conservative extension of T .

(b) Tc+eq is a conservative extension of Tc These two fact will be important to the following steps.

7. We suppose that there is a quasi-tautology β1 ∨ β2 ... ∨ βq in LH, with βi being a instance of the matrix αH . We now do a procedure of replacement of functions introduced for αH by constants in Lc. More speciﬁcally, if αH is of the form

1 If c1 and c2 are special constant for the formulas ∃xα1(x) and ∃xα2(x), then the equivalence axiom for these constants is ∀x(α1(x) ↔ α2(x)) → c1 = c2.

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θ[x, z1, f1(z1),..., zk, fk(z1,..., zk)]

and βi if of the form

θ[t, u1, f1(u1),..., uk, fk(u1,..., uk)],

we deﬁne the sequence of special constants d(u1), d(u1, u2),..., d(u1,..., uk) for eliminating the functions in the following manner:

Notation 10 .

(a) z i→ is a sequence of terms of the sequence z from the index i onward; (b) z i→j is a sequence of terms of z from the index i up to the index j;

The constant (d(u1))i is a special constant for the formula

(i+1)→ ∃(y1)i(¬∀y1 ∃z2∀y2 ... ∃zk∀yk 1→(i−1) i→ θ[t, u1, d(u1) , f1(z1) , ∗]), (1) begin ∗ a abbreviation that indicates the remain- ing sequence u2, f2(u1),..., uk, fk(u1,..., uk).

And, generally, (d(u1, u2,..., ui))j is a special constant for the formula

(j+1)→ ∃(yi)j(¬∀yi ∃zi+1∀yi+1 ... ∃zk∀yk 1→(j−1) j→ θ[t, ∗, d(u1,..., ui) , f1(zi) , ∗]). (2)

By successively applying the substitution axiom 0 and modus ponens, we have that, se βi is the formula

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θ[t, u1, d1(u1),..., uk, dk(u1,..., uk)],

0 then `Tc+eq βi → α. 8. Note that variables can occur in u. However, to ensure the use of the equivalence axiom properties in Tc+eq, we need all variables to be replaced by special constants. This will be necessary for the completion of the proof. So we make a second transformation in the quasi- tautology. Here equivalence axioms play an important role: they ensure that any two equivalent formulas refer to a single constant 2. With the addition of only the special axioms, we do not have this guarantee, since we could have two distinct elements satisfying the same formula of the form ∃xα. We introduce distinct special constants for each variable in u, obtaining u0. Subsequently, we prove, using the special equivalence axioms, that

0 0 `Tc+eq u1 = u1 → d(u1) = d(u1 ) 0 0 0 0 `Tc+eq u1 = u1 ∧ u2 = u2 → d(u1, u2) = d(u1 , u2 ) . . 0 0 `Tc+eq u1 = u1 ∧ uk = uk → 0 0 d(u1,..., uk) = d(u1 ,..., uk )

9. We use a similar procedure as in [5, p. 55] to 00 00 00 00 obtain a formula β1 ∨ β2 ... ∨ βq , being βi the

2Realizing this characteristic was instrumental in establishing the relationship between the techniques used in [9] and the techniques presented in the book Grundlagen der Mathematik [10]. In this paper, prior to his textbook of mathematical logic [5], Shoenﬁeld makes use of epsilon Hilbert’s calculus as a way of guaranteeing the uniqueness of the special constants. Mod- ern techniques make the conservative introduction of equivalence axioms.

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replacement of the functions and variables introduced by the special constants shown above. Recall that the second transformation ensure us 00 that `Tc+eq βi → α by the same procedure as in item 7.

We suppose C1,C2,...,Cm to be the proof sequence for the quasi-tautology that uses only identity and equality in T . Thus we prove that the transformation that have led β1 ∨ β2 ... ∨ βk 00 00 00 to become β1 ∨ β2 ... ∨ βk preserves tautologi- 00 00 00 cally the sequence. Therefore, β1 ∨ β2 ... ∨ βk is 00 00 00 tautological consequence of C1 ,C2 ,...,Cm. 00 It remains to prove that each Ci is a theorem of Tc+eq, when Ci is an axiom of T . This will result 00 00 00 in the proof of β1 ∨ β2 ... ∨ βk in Tc+eq. Each Ci is an axiom of identity, equality, an instance of identity or an instance of equality. We note 00 that, when we get Ci , we have transformed Ci into axioms of identity or equality, unless Ci is an instance of identity. This last case, though, is easily proved in Tc+eq from the propositions shown in the end of last item.

00 00 00 00 10. As Tc+eq ` β1 ∨ β2 ... ∨ βk and Tc+eq ` βi → α for each i, then Tc+eq ` α.

11. Since Tc+eq is a conservative extension of the logic in L, we lastly obtain that T ` α.

3.1 Finitary proof of equiconsistency In order to understand the ﬁnitary proof, it is important to consider procedures 7 and 8 of the previous section. In them, syntactic transformations are per- 00 formed to provide the formulas βi , eliminating the functions introduced to obtain the Herbrand’s normal formula.

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Each introduction of a function to eliminate a universal quantification is performed independently of the other introductions. In this sense, we can restrict the elimination procedure to certain universal quantifications. In fact, we can eliminate one, some or all functions introduced to obtain the Herbrand’s normal form. For the proof of the equiconsistency theorem, we will use the direct part of Herbrand’s theorem to obtain the quasi-tautology. Subsequently, we will make the procedure of the converse proof of Herbrand’s theorem restricted to the variables limited to sets. Before performing this procedure, we will eliminate all universal quantifications that are not restricted to sets in the NBG axioms. We will therefore prove that unrestricted universal quantifications are easily eliminated from the axiomatization presented.

Proposition 11 (Extensionality)

` (∀z ∈ V (z ∈ x ↔ z ∈ y) → x = y) ↔ (∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y)).

Proof. By substitution, we have

Σ =def ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y) {Σ} ` ∀z(z ∈ x ↔ z ∈ y) → x = y. And, by generalization,

Π =def ∀z(z ∈ x ↔ z ∈ y) → x = y {Π} ` ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y) We should now prove that

` Π ↔ (∀z ∈ V (z ∈ x ↔ z ∈ y) → x = y),

We know that (α → β ∧¬α → β) → θ is tautologically equivalent to β → θ. Thus, ∀z(z ∈ x ↔ z ∈ y) → x =

Manuscrito – Rev. Int. Fil. Campinas, v. 42, n. 2, pp. 85-112, Abr.-Jun. 2019. Alfredo Roque Freire and Rodrigo de Alvarenga Freire 98 y is tautologically equivalent to (∀z ∈ V (z ∈ x ↔ z ∈ y) ∧ ∀z∈ / V (z ∈ x ↔ z ∈ y)) → x = y. Nevertheless, since we know that, by deﬁnition, z ∈ V ↔ ∃w(z ∈ w), then z∈ / V is equivalent to ∀w(z∈ / w). From this, we have that ∀z∈ / V (z ∈ x ↔ z ∈ y) is a tautology. We thus obtain:

∀z(z ∈ x ↔ z ∈ y) → x = y

If, and only if,

(∀z ∈ V (z ∈ x ↔ z ∈ y)∧∀z∈ / V (z ∈ x ↔ z ∈ y)) → x = y

If, and only if,

∀z ∈ V (z ∈ x ↔ z ∈ y) → x = y

Note that we can use a similar procedure to limit other quantifications to sets, whenever we have a universal quantifier ∀z for a formula in which z ∈ x oc- curs. Although this technique does not work in all cases, for the axiomatization of NBG used in this article the procedure is effective. We will not expose here the corresponding proof for each axiom, for they are all very similar. In this sense, we have the following theorem:

Theorem 12 Let α be an axiom of NBG, then there is a formula α0 such that α0 is the elimination or restriction (to V ) of all universal quantiﬁcations occurring in α and ` α0 ↔ α.

We modify NBG yet one more time. The unrestricted existential quantiﬁers can be eliminated from instances of the scheme axiom for classes. We add, for each instance ∀v ∈ V ∃z∀x(x ∈ z ↔ (x ∈ V ∧α(x, v))), the constant cα and replace the axiom for ∀v ∈ V ∀x ∈ V (x ∈ cα ↔ (x ∈ V ∧ α(x, v))). Subsequently, we replace all other axiom of NBG by the version obtained

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Theorem 13 For every formula γ, if NBG ` γ, then U ` γ.

Proof. In order to proof this theorem, we should just treat the case in which γ is the scheme axiom for classes. From this and from theorem 12, we obtain easily that U proves all other axioms of NBG. We take a formula α with n free variables and in which all quantiﬁcations are bouded to V . We then prove that

U ` ∀v ∈ V ∃z∀x(x ∈ z ↔ (x ∈ V ∧ α(x, v))) (3)

(We call this formula θ). By replacement and generalization, we can eliminate the universal quantiﬁcations from the scheme axiom for classes. Thus we have U ` θ if, and only if,

U ` (v ∈ V ) → ∃z∀x(x ∈ z ↔ (x ∈ V ∧α(x, v))). (4)

Since x and z do not occur in the left side of the implication, then U ` θ if, and only if,

U ` ∃z∀x((v ∈ V ) (5) → (x ∈ z ↔ (x ∈ V ∧ α(x, v)))). Let θ0 be the formula

∀v ∈ V ∀x ∈ V (x ∈ cα ↔ (x ∈ V ∧ α(x, v))).

We now eliminate the quantiﬁers of θ0, obtaining

U ` ∀x((v ∈ V ) → (x ∈ cα ↔ (x ∈ V ∧ α(x, v)))). Thus, by replacement and modus ponens,

U ` ∃z∀x((v ∈ V ) → (x ∈ z ↔ (x ∈ V ∧ α(x, v)))).

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Therefore, U ` θ. We continue by proving the lemma:

Lemma 14 Let α be a sentence without variables for classes. If α is a theorem of U, then there is a proof of α in U that is free of unrestricted quantiﬁcations.

Proof. By hypothesis, we have a proof of α in U. Let γ [x] be the conjunction of axioms used in the given proof, in which x is the sequence of variables for classes that occur in the axioms. In this case, be the reduction theorem [5, p. 42]: T ` ∀xγ[x] → α, or, equivalently, T ` ∃x(γ[x] → α), for x does not occur in α, in which T is the theory without non logical axioms in the language of U. Let θ[x] be a prenex form of γ[x] → α. The formula θ[x] is of the form

∃z1∀y1...∃zk∀ykβ[x, z1, y1, ..., zk, yk]. From Herbrand’s theorem provability equivalence,

T ` ∃xθ[x] if, and only if,

TH ` ∃x∃z1...∃zkβ[x, z1, f1(x, z1), ..., zk, fk(x, z1, ..., zk)], in which TH is the theory obtained by the addition of function symbols in T , according to Herbrand’s normal form ∃xθ[x]. Let β0 be a open formula

β[x, z1, f1(x, z1), ..., zk, fk(x, z1, ..., zk)].

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By Herbrand’s theorem, there is a quasi-tautology 0 0 0 0 β1 ∨ ... ∨ βm, in which each βi is an instance of β in the language of TH . Let’s build the appropriate proof of α in Uc+eq, obtained from U by the addition of special constants, special axioms and special axioms of equivalence. 0 From start, we replace each free variaable in β1∨...∨ 0 βm for special constants. The result is a disjunction of m sentences that are quasi-tautological in Uc+eq. This quasi-tautology is the starting point of the proof of α in Uc+eq.

Subsequently, we replace the occurences of f1(a, b1), ..., fk(a, b1, ..., bk) for sequences of appropriate special constants. For this, we follow the items 7 and 8 in the proof of Herbrand’s theorem. 0 0 The result is the disjunction β1c ∨ ... ∨ βmc, tautological consequence of instances of identity, equality, special axioms and special equivalence axioms, having quantiﬁcations only for the variables z1, y1, ..., zk, yk. Let θic be the formula

∃z1∀y1...∃zk∀ykβ[ti, z1, y1, ..., zk, yk], obtained from βic by the restating the quantifications ∃z1∀y1...∃zk∀yk. The closed terms in ti are in Uc+eq. 0 Each disjunction θic implies the corresponding βic in Uc+eq, as we have seen in the first paragraph of item 9 in Herbrand’s theorem proof. To show this, we use only simple properties of implication, of the quantifiers ∃z1...∃zk and the special axioms used in the replacement of the occurrences f1(a, b1),...,fk(a, b1, ..., bk) described above. From what we have exposed, it follows that

Uc+eq ` θ1c ∨ ... ∨ θmc, without using quantiﬁcations over classes. However, since θ[x] is a prenex form of γ[x] → α, we have

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Uc+eq ` θ[x] ↔ (γ[x] → α), using only variations of quantiﬁcations that occur in γ[x] → α. Therefore, no quantiﬁcation for class variables. On the other hand, by the rule of substitution,

Uc+eq ` θ[ti] ↔ (γ[ti] → α).

As θ[ti] é θic, we conclude, using tautological consequence, that

Uc+eq ` γ[t1] ∧ ... ∧ γ[tm] → α, without using quantiﬁcation for class variables. On the other hand, γ[x] is the conjunction of axioms in Uc+eq. Each γ[t1] can be proved in Uc+eq using only tautological consequences and instances of the substitution rule. Therefore,

Uc+eq ` α, without using quantification for class variables. Finally, we observe that any proof of α of U in Uc+eq can be transformed in a proof in U of the same α, and that this transformation introduces only quantifications directly related to the special axioms used [5, p. 52]. Since we haven’t used special axioms for class variables, the result follows. From this result, we prove by finitary means the equiconsistency result:

Theorem 15 Let α be a sentence with all its quanti- ﬁers bounded to sets and such that NBG ` α. Then, ZF ` α.

Proof. Let α be a sentence with all its quantiﬁers bounded to sets and such that NBG ` α. Thus, by lemma 14, there is a proof in U in which no unbounded quantiﬁcations occur. We call this proof sequence Seq.

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We make transformations in this proof sequence that preserve tautological consequences, do not aﬀect α and such that the transformed axioms are theorems of ZF. Let x1, x2, . . . , xk be the free variables that occur in Seq. We add the following initial segment to the proof ∗1 sequence x1 ∈ V, x2 ∈ V, . . . , xk ∈ V , obtaining Seq . Next, we apply the transformation ∗ZF in the formulas in Seq∗1 . Every occurrence

1. cθ = cα, are replaced by ∀y ∈ V (y ∈ cθ ↔ y ∈ cα)

2. cθ ∈ x, are replaced by ∃y ∈ V (y = cθ ∧ y ∈ x)

3. cθ ∈ cα, are replaced by ∃y ∈ V (y = cθ ∧ y ∈ cα)

4. x = cα, are replaced by ∀y ∈ V (y ∈ x ↔ y ∈ cα)

5. x ∈ cθ, are replaced by θ(x)

The successive application of this transformation ∗1 to Seq eliminate all occurrences of cθ, forming the sequence Seq∗2 . ∗ Recall that, for each logical axiom Axiomj in Seq 1 , ∗ZF we should verify whether Axiomj is also a logical axiom or a consequence of ZF together with formulas x1 ∈ V, x2 ∈ V, . . . , xk ∈ V . In the second case, we ∗ZF ∗2 replace each axiom Axiomj in Seq by the proof ∗ZF ∗3 sequence of Axiomj . Hence, we obtain Seq . We know the following proposition about functors [5, p. 30]:

Proposition 16 A functor ∗ of formulas to formulas satisﬁes for every formula α and β:

1. (¬α)∗ é ¬α∗

2. (α ∨ β)∗ é α∗ ∨ β∗

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Thus, if δ is tautological consequence of γ1, γ2, . . . , γn, ∗ ∗ ∗ ∗ then δ is tautological consequence of γ1 , γ2 , . . . , γn.

Therefore, all transformations described above do not aﬀect the proof tautologically. However, some instances of the logical axioms may not be logical axioms after the transformation. We now investigate what happen with logical axioms in which constants of U occur.

1. Substitution axiom

There is no instance of the substitution θx(c) → ∃xθ in Seq∗1 since unrestricted quantiﬁcations do not occur in the initial proof in U.

2. Identity axiom ∗ZF Note that (cα = cα) is ∀y ∈ V (α(y) ↔ α(y)). And this last one is a tautology.

3. Equality axiom

(a) If the axiom is of the form x1 = cα ∧ x2 = y2 → x1 ∈ x2 ↔ cα ∈ y2, then, after the transformation:

(∀z ∈ V (z ∈ x1 ↔ α(z)) ∧ x2 = y2) →

(x1 ∈ x2 ↔

∃w ∈ V (∀z ∈ V (z ∈ w ↔ α(z)) ∧ w ∈ y2))

We show that this formula is theorem of ZF and of the formulas x1 ∈ V, x2 ∈ V, . . . , xk ∈ V .

If we have that ∀z ∈ V (z ∈ x1 ↔ α(z)) ∧ x2 = y2 and we suppose that x1 ∈ x2, the, by corollary 2 of the identity theorem in [5, p. 36], we obtain

∀z ∈ V (z ∈ x1 ↔ α(z)) ∧ x1 ∈ y2.

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Then, by the substitution axiom in e x1 ∈ V ,

∃x1 ∈ V (∀z ∈ V (z ∈ x1 ↔ α(z))∧x1 ∈ y2). Using the variant theorem [5, p. 35],

∃w ∈ V (∀z ∈ V (z ∈ w ↔ α(z)) ∧ w ∈ y2).

On the other hand, if we suppose that ∃w ∈ V (∀z ∈ V (z ∈ w ↔ α(z)) ∧ w ∈ y2), then, since ∀z ∈ V (z ∈ x1 ↔ α(z)) and by extensionality in ZF, we prove that ∃w ∈ V (w = x1∧w ∈ y2). Hence, we obtain that x1 ∈ y2. As, from hypothesis, x2 = y2, we conclude that x1 ∈ x2, ﬁnalizing the proof.

(b) x1 = y1 ∧ cα = y2 → x1 ∈ cα ↔ y1 ∈ y2. The strategy for item b is similar to item a.

(c) x1 = y1 ∧ cα = cβ → x1 ∈ cα ↔ y1 ∈ cβ. For this case, we obtain from the transformation: x1 = y1 ∧∀z ∈ V (α(z) ↔ β(z)) → α(x1) ↔ β(y1). As x1 and y1 are free variables in the formula, the transformed formula is tautological consequence of ∀x1 ∈ V ∀x2 ∈ V and instances of identity axioms.

(d) cβ = cα ∧ cγ = y2 → cβ ∈ cγ ↔ cα ∈ y2.

(e) cβ = cα ∧ x2 = y2 → cβ ∈ x2 ↔ cα ∈ y2.

(f) cβ = cα ∧ cγ = cψ → cβ ∈ cγ ↔ cα ∈ cψ.

We prove items d, e and f by a simple combina- tion of the strategies used in a and c.

Let’s now evaluate what occur to the aximos in U after the transformation ∗ZF : The instances of extensionality, scheme for classes, replacement for classes and foundations are the only

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ones we aﬀect by applying ∗ZF (only in those axioms constants of U occur).

1. Extensionality. We evaluate two cases:

∀y ∈ V (y ∈ x ↔ y ∈ cθ) → x = cθ.

∀y ∈ V (y ∈ cα ↔ y ∈ cθ) → cα = cθ.

After ∗ZF , we have respectively ∀y ∈ V (y ∈ x ↔ θ(x)) → ∀y ∈ V (y ∈ x ↔ θ(x)) ∀y ∈ V (α(x) ↔ θ(x)) → ∀y ∈ V (α(x) ↔ θ(x)). Both are tautologies.

2. Scheme for classes. Let v1, v2, . . . , vn be free variables occurring in θ

∀v1v2 . . . vn ∈ V ∀y ∈ V (y ∈ cθ ↔ θ(y))

After ∗ZF , we have

∀v1v2 . . . vn ∈ V ∀y ∈ V (θ(y) ↔ θ(y)) And this is a tautology.

3. Replacement for classes.

∀x ∈ V (func(cθ) → ∃y ∈ V ∀w(w ∈ y ↔ ∃v ∈ x((v, w) ∈ cθ))) becomes

∀x ∈ V (∀v1v2v3 ∈ V (θ(v1, v2) ∧ θ(v1, v3) → v2 = v3) → ∃y ∈ V ∀w1(w1 ∈ y ↔ ∃w2 ∈ x(θ(w2, w1)))) Which is precisely the replacement axiom for ZF.

4. Foundation.

(cθ 6= ∅ → ∃y ∈ V (y ∈ cθ → cθ ∩ y = ∅)) becomes ∃x ∈ V θ(x) → ∃y ∈ V (θ(y) → ∀w(w∈ / y ∨ ¬θ(w)));

We suppose ∃x ∈ V θ(x). Then ` ∃x(x ∈ Va ∧ θ(x)), for some ordinal a. Let A = {x|x ∈ Va ∧

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θ(x)}, it follows that ZF ` A 6= ∅. By the axiom of foundation, ∃y(y ∈ A → A ∩ y = ∅). This is equivalent to

∃y(θ(y)∧y ∈ Va → ∀w¬(w ∈ y ∧θ(w)∧w ∈ Va))

(as w ∈ y → w ∈ Va)

∃y(θ(y) ∧ y ∈ Va → ∀w¬(w ∈ y ∧ θ(w)))

That is, the formula is a theorem of ZF.

When axioms of U occur without constants, they can be understood as axioms of ZF, since we have added the formulas xi ∈ V .

1. Extensionality ∀y ∈ V (y ∈ x ↔ y ∈ z) → x = z, since we have x ∈ V and z ∈ V , represents extensionality in ZF.

2. The same is true for the axiom of replacement and foundation.

3. The axiom scheme for classes do not occur without constants in U.

Therefore, in the sequence Seq∗3 , we have:

1. formulas of the form x ∈ V ,

2. logical axioms,

3. axioms of ZF,

4. and all others are consequence of logical infer- ences from previous formulas in the sequence.

This is a proof in ZF. Since no transformation aﬀect α, we have proved that ZF ` α.

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Corollary 17 If there is α such that NBG ` α ∧ ¬α, then there is a procedure that generate β such that ZF ` β ∧ ¬β.

Proof. If NBG ` α ∧ ¬α, then NBG proves any formula. In particular, it proves a formula β¬β in which all quantiﬁcations are bounded to sets. Thus, by the theorem, ZF ` β ∧ ¬β.

4 There is no interpretation of NBG in ZF

Before we prove there is no interpretation os NBG in ZF, we show some deﬁnitions as propositions:

Deﬁnition 18 Let V be a model of ZF and M a class V -deﬁnable, we say that the model M = (M, ∈V ) is a V -natural model.

Deﬁnition 19 Let M be a model in the language LZF (the only non logical symbol is membership) and an I interpretation I = hU, φi of LZF in LZF (we write ∈ for φ(∈)), then we deﬁne the model MI = (A, ∈IM ) em L as

A = {x | M U(x)} and IM I ∈ = {hx, yi | M U(x) ∧ U(y) → x ∈ y} From this deﬁnition, we can easily prove by induction that:

Proposition 20 Let M be a model in LZF and I = I hU, ∈ i be an interpretation of LZF in LZF , then, for all sentences α

I I M α ⇐⇒ M α The following proposition is a strengthening of a result in [11]. They show that if the existence of a transitive model of ZF is consistent with ZF, then there

Manuscrito – Rev. Int. Fil. Campinas, v. 42, n. 2, pp. 85-112, Abr.-Jun. 2019. The Ontological Import of Adding Proper Classes 109 cannot be a set-interpretation of ZF in ZF. Here, we replace that consistency condition for “ZF does not prove the inconsistent sentence for ZF itself”.

Proposition 21 Let I = hU, ∈I i be an interpretation of ZF in ZF, if ZF ` {x | U(x)} is a set, then ZF ` ¬Con(ZF ).

Proof. Take I as in the proposition and suppose ZF ` {x | U(x)} is a set. Let pαq be the Gödel number of the formula α rep- I resented in ZF. Since M is a set for every M ZF , we deﬁne recursively the set T for each M ZF : Notation 22 a([k] = b) is the replacement of the k’th element of the sequence a for b.

I hpαq, ai ∈ T if, and only if, a ∈ M and M I (1) if α if atomic of the form xi ∈ xj, hai, aji ∈ (∈ ) (2) if α is of the form β ∧ γ : hpβq, ai ∈ T and hpγq, ai ∈ T (3) if α is of the form ¬β : hpβq, ai ∈/ T (4) if α is of the form ¬∃xkβ : hpβq, a([k] = b)i ∈ T for some b ∈ M I

By ﬁnite induction over the formula complexity, we prove that

I M hpϕq, ai ∈ T if, and only if, M ϕ(a)

Take P rZF (x, y) to be the provability predicate for ZF deﬁned in ZF and representing the statement “x is the number of the proof y”. Then, we say that I T h(ZF ) = {y | ∃xP rZF (x, y)}. Hence, since M I ZF , M T h(ZF ) ⊆ T . As M 2 ∅ ∈ ∅, we have p∅ ∈ ∅q ∈/ T . From this, we obtain M p∅ ∈ ∅q ∈/ T h(ZF ). Once M is arbitrary, by the completeness theorem, if ZF has a model, then ZF ` p∅ ∈ ∅q ∈/ T h(ZF ).

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This is an absurd by Gödel’s incompleteness theorem. Thus, ZF ` ¬Con(ZF ). Deﬁnition 23 Let V be a model of ZF and M a V - natural model. We say that M reﬂect a formula ϕ(x) if, and only if, for every a ∈ M

V ϕ(a) ⇐⇒ M ϕ(a) Theorem 24 [Reflection theorem] [6, p. 168] Let V be a model of ZF e and φ1, φ2, . . . , φn any sequence of formulas, then, there is an ordinal a such that M = (Va, ∈) reflect φi for i between 1 and n. We show that the desired result is a consequence of the reflection theorem and the fact that NBG is finitely axiomatizable:

Theorem 25 There is no interpretation of NBG in ZF.

Proof. Suppose there is an interpretation I of NBG in ZF. Since the number of axioms in NBG can be said to be ﬁnite, there is a formula α that is equivalent to the conjunction of all NBG’s axioms:

α is Axiom1 ∧ Axiom2 ∧ ... ∧ Axiomn. Thus, NBG ` α and, from the interpretation theorem for first-order logic, ZF ` αI . I Suppose V ZF , then V α . By the reflection I theorem, there is an ordinal a such that Va α . It I follows that Va α. I Since Va is a set, we obtain that the domain in Va is also a set. ∗ We define the model V in LZF :

∗ I 1. The domain D in V is such that D = {x | Va ∃y(x ∈ y)}.

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V ∗ I 2. The predicate ∈ = {hx, yi | Va x ∈ y}.

Since NBG proves the restriction to sets of all ZF ∗ axioms, we have that V ZF . Since the domain I Va is a set, it follows that D is also a set. It means that ZF ` Con(ZF ), absurd by the incompleteness theorem.

4 References

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[9] Shoenfield, J.R. “A relative consistency proof” The Journal of Symbolic Logic 19, 4, pp. 21-28, 1954.

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